Space time curvature affects the orbits of many spacecraft. Think of a space craft as a marble; now think of the space time plane as a bowl with an orange in it (the orange is the sun). If you roll the marble into the bowl it will spiral around until it hits the orange, if you roll it too fast it will escape the suns gravity and fly out of the bowl. If you roll the marble with exactly the correct speed it will roll around and around and never stop: this is an orbit. Now if you were to poke the marble as it rolled around continuously you would send the marble either out of the bowl or into the orange. This is how space crafts re-enter the atmosphere (by slowing down). On a human scale, space time does not matter much, but on the scale of stars and other large, massive features it matters more.